Spatiotemporal imaging and shaping of electron wave functions using novel attoclock interferometry

Electrons detached from atoms by photoionization carry valuable information about light-atom interactions. Characterizing and shaping the electron wave function on its natural timescale is of paramount importance for understanding and controlling ultrafast electron dynamics in atoms, molecules and condensed matter. Here we propose a novel attoclock interferometry to shape and image the electron wave function in atomic photoionization. Using a combination of a strong circularly polarized second harmonic and a weak linearly polarized fundamental field, we spatiotemporally modulate the atomic potential barrier and shape the electron wave functions, which are mapped into a temporal interferometry. By analyzing the two-color phase-resolved and angle-resolved photoelectron interference, we are able to reconstruct the spatiotemporal evolution of the shaping on the amplitude and phase of electron wave function in momentum space within the optical cycle, from which we identify the quantum nature of strong-field ionization and reveal the effect of the spatiotemporal properties of atomic potential on the departing electron. This study provides a new approach for spatiotemporal shaping and imaging of electron wave function in intense light-matter interactions and holds great potential for resolving ultrafast electronic dynamics in molecules, solids, and liquids.

Importantly, one can notice that, the interference patterns calculated by SFA model in the cases of θ=0°and 90° (Supplementary Fig. 2e and 2g) are basically consistent with that calculated by the CCSFA model (Supplementary Fig. 2i and 2k) if ignoring the amplitude modulation induced by the Coulomb attraction.This implies that in these two specific cases the Coulomb effect has tiny impact on the phase of electron wave packet and thus can be neglected in the following analysis.Supplementary Fig. 2| Two

IV. Derivation of interference formula in the employed two-color fields
As illustrated in the main text, the interference pattern in two-color fields can be well accounted for by the interference of four electron wave packets released in two consecutive 800 nm cycles, i.e., 2 1 2 3 4 ( , ) as expressed in Eq. ( 1).

S(1)
Since 800 nm is perturbative weak, we assume it has tiny effect on the ionization instants of the electron wave packets ψ1 and ψ2 and the pre-exponential factor ρs(p).
Therefore, ts2 =ts1+T400, and ρs1(p)≈ρs2(p).In the following, we label ρs1(p) and ρs2(p) as ρs(p).Correspondingly, the complex phases of the two electron wave packets, i.e., ψ1 and ψ2, can be expressed as is a highorder small quantity that depends on the laser intensity of 800 nm field, therefore in the following derivation we can neglect its contribution.By substituting the phases in Eq.S(2) into Eq.S(1), one can obtain the following formula: From this formula, one can see that the phase information (i.e., Re[S10]) of the unperturbed electron wave function is lost when considering the interference effect.
Further, we deduce the analytical formula of σ according to its As ts2=ts1+T400, we find σ2=-σ1.This indicates that the two electron wave packets emitted from adjacent 400 nm cycles experience opposite amplitude and phase modulations.In the following, we denote σ1 as -σ and σ2 as σ.Then, we rearrange the interference formula as: Note that b=(Up (400) +Up (800) +Ek+Ip)T800, we can further arrange it into the formula of b=2EkT400+2a, with a=(Up (400) +Up (800) +Ip)T400.Therefore, the interference formula in the two-color fields can be finally expressed as

(
-color phase-resolved photoelectron energy spectra at different emission angles.a-d Experimental results.e-h SFA calculations.i-l CCSFA calculations.The polarization configurations of two-color field vectors are labeled on the top.III.Amplitude modulation induced by the change of ρ(p) when adding a weak 800 nm fieldBased on the SFA model within saddle-point approach, we can directly calculate the pre-exponential factor 0 in single-and two-color fields.Here, we label them as 0 () s  p and 1 () s  p , respectively.Then, we characterize the amplitude modulation induced by the change of pre-exponential factor using of δ for parallel interaction configuration is shown in Supplementary Fig.3a.For comparison, we also present the amplitude modulation induced by the change of imaginary part of the complex phase., S0 and S represent the complex phase of electrons accumulated in single-and two-color fields.The corresponding result is shown in Supplementary Fig.3b.One can see that the amplitude modulation induced by ρ(p) exhibits similar phase and energy dependences as that induced by the change of imaginary part of the complex phase.However, the magnitude of amplitude modulation induced by ρ(p) is quite smaller as compared with that induced by the change of imaginary part of complex phase.Therefore, in the following analysis we can neglect the amplitude modulation contribution induced by ρ(p) when adding a weak 800 nm field.Supplementary Fig. 3| Calculated amplitude modulation δ of electron wave packet when adding a weak linearly polarized 800 nm field in parallel interaction configuration.a Amplitude modulation induced by the change of ρ(p).b Amplitude modulation induced by the change of imaginary part of the complex phase.
− = pand Re[S10] to denote the amplitude and phase of the unperturbed electron wave function in 400 nm circular fields.Accordingly, the interference formula can be arranged into: